Applied Mathematical Sciences, Vol. 10, 2016, no. 64, 3143 - 3164

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2016.69248

Application of the Statistical Physics Methods

for the Investigation of Phase Transitions

in the Lotka–Volterra System with

Spatially Distributed Parameters

Yu. V. Bibik

Dorodnicyn Computing Center, Federal Research Center “Computer Science and

Control” of Russian Academy of Sciences

Vavilov str. 40, 119333, Moscow, Russia

Copyright © 2016 Yu. V. Bibik. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

The application of statistical physics methods for the investigation of phase

transitions for a generalization of the Lotka–Volterra system with additional terms

that take into account the spatial distribution of species is considered. The

additional terms are chosen in such a way that the generalized system, as well as

the classical Lotka–Volterra system, is Hamiltonian. Kerner's approach, who was

the first to use statistical mechanics for the analysis of Hamiltonian Volterra-type

systems, is used to investigate the generalized Lotka–Volterra system. In addition,

the Lee–Yang approach is used to calculate zeros of the statistical sum. The

position of these zeros is used to find out whether there are phase transitions in the

model under examination.

Keywords: Lotka–Volterra system, statistical sum, gamma function, Lee–Yang

zeros

1. Introduction

It has been shown (e.g., see [4], [22], [25], [26], [28], [29], [30] [31]) that in some ecological and biological systems processes similar to phase transitions in statistical

3144 Yu. V. Bibik

physics are observed. In such systems, qualitative changes occur similarly to the

statistical and thermodynamic changes that are observed in phase transitions in

statistical physics. Examples of such processes that have not yet found a proper

scientific explanation are as follows:

– Qualitative changes in forest ecosystems in the course of ecological succession

when, e.g., a deciduous forest is replaced by coniferous forest and (or) a

community of herbaceous plants is replaced by a community of woody plants.

– Bursts of mass reproduction of lemmings in tundra zone accompanied by their

migration from the habitat and mass mortality. As a result, the population of

lemmings first significantly decreases and then gradually increases up to the new

burst of mass reproduction.

– Periodically repeating bursts of mass reproduction of forest insects.

– Phase transitions in biological membranes accompanied by the transition from

the liquid crystal state to the solid crystal state and vice versa.

In this paper, we apply methods and approaches of statistical physics for the

analysis of phase transitions in ecological and biological systems. The choice of

statistical physics methods and approaches is explained by the fact that during

many years of the development of statistical physics, a vast amount of

experimental and theoretical data that make it possible to solve complicated

problems have been accumulated.

The theory of phase transitions is in important branch of statistical physics. The

theory of phase transitions and related critical phenomena has a rich history. Many prominent physicists, both theorists and experimenters, contributed to the

development of the theory of phase transitions. Even though the first significant

studies of critical phenomena in various substances appeared as early as in the

second part of the 19th century, the development of this division of the statistical

physics continues.

Studies go on in various divisions of physics, and the accumulated data are

steadily updated. The mathematical methods are elaborated, and advanced devices

are used that improve the accuracy of experiments.

Presently, the modern statistical physics has a variety of methods,

approaches, and theories that make it possible to describe, analyze, and

understand the essence of the processes and critical phenomena that occur

during phase transitions in various systems (see [1], [2], [6], [7], [8], [9], [12],

[17], [20], [21], [23], [27], [43], [48], [49], [50], [52], [54], [55], [56], [57]).

The large number of methods, approaches, and theories is explained by the

complexity and diversity of phase transitions and critical phenomena that cannot

be explained using a unified theory or approach.

The first theory used a hundred years ago to theoretically describe critical

phenomena was the mean field theory.

Among the mean field theories are the Gibbs critical point theory [18], [19], the

Weiss theory of ferromagnetism, and the Landau theory of phase transitions of the

second kind [44], [45].

The mean field theory turned out to be inapplicable when fluctuations become

significant, i.e., in the vicinity of the point of phase transition.

Application of the statistical physics methods for the investigation… 3145

The fluctuation theory of phase transitions of the second kind [53] describes the

interaction of fluctuations, makes it possible to take into account the interaction of

fluctuation phenomena at critical points, and determine the values of critical

parameters.

The microscopic theory of phase transitions makes it possible to investigate the

properties of concrete models that undergo a phase transition. The main difficulty

in the development of the microscopic theory of phase transitions is the absence

of an explicit small parameter.

Recently, quantum phase transitions have been widely studied (see [24], [41],

[42], [71]); transitions of matter from one quantum thermodynamic phase into

another under changes of the external conditions in the case when there are no

thermal fluctuations are investigated.

An important step in the history of the development of phase transition theories

are the theories that appeared in the 1960s and 1970s, which enriched the

statistical physics by the concepts of renormalization group, scale invariance, and

universality (see [10], [11], [13], [14], [15], [16], [33], [34], [35], [36], [64],

[65], [66], [67], [68], [69], [72]).

Using the coarsening techniques, these methods provide a tool for the analysis of

such difficult physical problems as quantum field theory and the theory of critical

phenomena.

The renormgroup divides the ferromagnetic system in the vicinity of a critical

point into cells of which each contains a small number of atomic-level magnets;

the size of cell determines the scale. This method made it possible to describe the

behavior in the vicinity of the critical point and obtain a quantitative estimate of

the properties of the system under examination on a computer.

The scale invariance is the property of the equations describing a physical theory

or process to remain invariant when all the distances and time intervals are

multiplied by the same factor.

A large number of model systems are used in statistical physics to describe

and investigate phase transitions and critical phenomena. Such models include

lattice magnetic models, such as two-dimensional and three-dimensional Ising

models, Heisenberg's model, Matsubara and Matsuda models, Stanley n-

component models, spherical Berlin and Kac model, lattice percolation model

used to investigate complex systems, the lattice gas model for the description of

the liquid–vapor critical point, and others.

Among them, the Ising model, which is widely used for the investigation of phase

transitions, deserves special attention. This model was designed for the

investigation and description of ferromagnetic phenomena. Ising claimed in 1924

(see [32]) that there are no phase transitions in one-dimensional systems and the

one-dimensional lattice does not exhibit ferromagnetism.

In 1942, Onsager obtained an exact solution for the two-dimensional Ising model

[51].

One of the first researchers who applied the methods and approaches of

statistical physics for the investigation of biological models was the American physicist E.H. Kerner. Beginning from 1957, he published a series of papers [37],

3146 Yu. V. Bibik

[38], [39], [40] devoted to the application of statistical mechanics methods for the

analysis of Volterra-type models. Using the features of these systems, Kerner

developed for them analogs of the main thermodynamic parameters, such as

internal energy, entropy, and statistical sum. Kerner explicitly calculated the

statistical sum for the Lotka–Volterra equation. He also discovered a remarkable

feature of the Hamiltonian of the Lotka–Volterra system. This is the fact that the

statistical sum for this system can be calculated explicitly in terms of the gamma

functions up to a factor representable in terms of the elementary functions.

Kerner's studies confirmed the possibility and usefulness of applying the statistical

mechanics methods to the analysis of Volterra-type equations.

In addition to Kerner's studies, the methods of statistical physics were used for the

investigation of biological and ecological models in [3], [5], [58], [59], [60].

In this paper, we want

– to use the result of theoretical and experimental studies of phase transitions and

critical phenomena obtained in statistical physics for the investigation of phase

transitions in biological and ecological systems by finding an appropriate simple

mathematical biology model;

– to find a calculation method and use mathematical transformations and

approximations that would enable us to obtain the result as easily as possible

(practically manually) without using long and complicated computer calculations;

– to obtain an analytical solution that would enable us to determine graphically

whether there is a phase transition in the system.

The aim of this paper is to develop an approach that provides simple means for

determining whether there is a phase transition in the system.

For the investigation of qualitative changes in biological and ecological

systems that resemble phase transitions in physical systems, we use a

generalization of the classical Lotka–Volterra model [47], [61], [62], [63] in

which terms taking into account the spatial distribution of species are added.

As the investigation techniques, we use the approach of Kerner, who was the

first to apply statistical physics methods for the analysis of Volterra systems,

and the Lee–Yang approach [46], [70], which allows one to detect the existence

of phase transitions from the position of zeros on the complex plane.

The thermodynamic state and properties of the system are studied using the

statistical sum, which is represented in terms of a simple combination of

elementary and gamma functions. The existence and nonexistence of phase

transitions is determined based on the position of the zeros of the statistical sum

on the complex plane.

The paper is organized as follows:

1. Introduction.

2. The description of the investigation techniques and features of the model.

3. Calculation of the statistical sum for the model under examination.

4. Investigation of the phase transition in the model.

5. Conclusions.

Application of the statistical physics methods for the investigation… 3147

2. The description of the Investigation Techniques and Features of

the Model

2.1. The Investigation Technique

We use the approach of Kerner [3], [38], [39], [40], who was the first to apply

statistical mechanics methods for the analysis of Volterra systems. Kerner used a

change of variables that allowed him to calculate the statistical sum in terms of the

gamma functions accurate to elementary functions. Kerner applied the change of

variable directly to the whole statistical sum. In this paper, we cannot apply

Kerner’s approach directly because in our case we have extra terms added to the

classical Lotka–Volterra system that take into account the spatial distribution of

species. This considerably complicated the equations under examination. To

overcome this difficulty, we expand the statistical sum into Taylor’s series and

apply Kerner’s change of variable to each term of this series, which has a proper

form. In this case, each term is represented by a combination of the gamma

functions and elementary functions. Then, the series is summed to obtain the final

statistical sum.

The existence or nonexistence of phase transitions in the system is determined

graphically judging by the position of the zeros of the statistical sum on the

complex plane.

2.2. A Description of the Model Features

The classical Lotka–Volterra system has the form

dt

d,

dt

d. (2.2.1)

Here is the population of prey, and is the population of predators.

The classical Lotka–Volterra system does not take into account the dependence of

the species of predators and prey on spatial coordinates. Taking into account this

dependence can considerably improve the analytical power of the model. To take

the spatial dependence into account, we denote the flows of predators and prey by

1j and 2j , respectively. The generalization of the Lotka–Volterra system with

account for the spatial dependence of the predator and prey population has the

form

1divjdt

d

, (2.2.2)

2divjdt

d

. (2.2.3)

3148 Yu. V. Bibik

The contribution of the predator and prey population to time derivatives is

determined by the divergences of the flows 1j and 2j .

We assume that the flow 1j is equal to the gradient of the predator population

with the opposite sign:

1j . (2.2.4)

The interpretation is that the prey tend to the place where the population of

predators is lower. Similarly, we assume that the flow of predator population 2j

equals the gradient of the prey population :

2j . (2.2.5)

This means that the predators tend to the place where the prey is more numerous.

Under these assumptions, the generalization of the Lotka–Volterra system takes

the form

dt

d, (2.2.6)

dt

d. (2.2.7)

To be able to apply the statistical mechanics techniques to the above

generalization of the Lotka–Volterra system with account of the spatial

dependence, we transform Eqs. (2.2.6) and (2.2.7) to Hamiltonian form. Make the

change of variables lnq , lnp .

In the new variables, Eqs. (2.2.6) and (2.2.7) take the form

22 )()(1 y

p

yy

p

x

p

xx

pp pepepepeedt

dq , (2.2.8)

22 )()(1 y

q

yy

q

x

q

xx

qq qeqeqeqeedt

dp . (2.2.9)

We reduce the terms containing spatial derivatives appearing in Eqs. (2.2.8) and

(2.29) to Hamiltonian form. We assume that the values of the spatial derivatives

are small and, therefore, we can neglect the quadratic terms. In addition, the same

assumptions imply that the coefficients multiplying pe and qe may be replaced

by constants. Using appropriate linear transformations of the variables x and y ,

Application of the statistical physics methods for the investigation… 3149

we can make these constants equal to unity. Then, Eqs. (2.2.8) and (2.2.9) take the

form

yyxx

p ppedt

dq1 , (2.2.10)

yyxx

q qqedt

dp 1 . (2.2.11)

Equations (2.2.10) and (2.2.11) are Hamiltonian equations with the Hamiltonian

)2

1

2

1

2

1

2

1( 2222

yxyx

pq qqpppeqedxdyH . (2.2.12)

Unfortunately, the form of Hamiltonian (2.2.12) does not allow one to use

Kerner's approach because the statistical sum cannot be explicitly calculated in

terms of the gamma functions accurate to elementary functions. To resolve this

problem, we select new interaction terms that are similar to the interaction terms

in Hamiltonian (2.2.12). These new interaction terms should enable us to calculate

the statistical sum in terms of the gamma and elementary functions. We select a

replacement for the terms 2

xp and 2

yp in Hamiltonian (2.2.12) using linear and

exponential functions. An appropriate function to replace 2x is xe x . Instead of

the variable x , we will use xp and yp .

Upon the transformations, Hamiltonian (2.2.12) takes the form

)( yx

qqq

yx

ppp qqeeqeppeepedxdyH yxyx . (2.2.13)

After the integration, the total derivatives xp , yp ,

xq , and yq disappear. The

resulting expression for the Hamiltonian is

)( yxyxqqqppp eeqeeepedxdyH . (2.2.14)

For this Hamiltonian, the Hamiltonian equations take the form

yy

p

xx

pp pepeedt

dqxx 1 , (2.2.15)

yy

q

xx

qq qeqeedt

dpxx 1 . (2.2.16)

3150 Yu. V. Bibik

Due to the above assumption that xp , yp ,

xq , and yq are small, the quantities

,xpe yp

e and xqe , yq

e are close to unity. The Hamiltonian equations (2.2.15) and

(2.2.16) are close to the Hamiltonian equations (2.2.10), (2.2.11). This confirms

the fortunate selection of the new spatial terms. In the next section, we calculate

the statistical sum for Hamiltonian (2.2.14).

3. Calculation of the Statistical Sum for the Model under

Examination

In this section, we calculate the statistical sum for Hamiltonian (2.2.14) with the

new spatial terms. To this end, we replace the continuum of the points yx, with

the discrete set of points on which the periodic boundary conditions are given. It

suffices to select three points on each axis x and y . As a result, a square grid

consisting of nine points will be used for the calculations. The derivatives of the

dependent variables p and q with respect to x and y will be replaced by finite

differences. For simplicity, the grid size is set to unity. For convenience, we

rename q to and p to . Note that, due to the additivity of Hamiltonian

(2.2.14), the statistical sum to be calculated is represented as the product of two

identical statistical sums

21ZZZ . (3.1)

The statistical sum 1Z has the form

)1,,(

,

),1,(

,

,, )(

1 ][

jiji

ji

jiji

ji

jiji eee

eedZ

. (3.2)

The statistical sum 2Z has the same form as 1Z .

Following Kerner's method, we try to represent the expression for the statistical

sum by a combination of the gamma and elementary functions. However, the term

in (3.2) containing )( ,1, jijie prevents the representation of the integral of interest

as Euler's integral, which is the integral expression for the gamma function. To

overcome this difficulty, we make the following transformations in the expression

ji

jijijiji eeeE ,

1,,,1, )()(

(3.3)

in (3.2). Assume that the major contribution to the statistical sum is made by the

small values ji , . Then, we have the following heuristic consideration for the

transformation of jijie ,1, . Note that both 1, jie

and 1,1 jie

are small.

Therefore, it holds that

Application of the statistical physics methods for the investigation… 3151

....)1()1(1

)]1(1)][1(1[

,1,

,1,,1,,1,

jiji

jijijijijiji

ee

eeeee

. (3.4)

Next, we replace ji ,1 by 1,1 jie

to obtain

ji

jiji

jijiji ee eeeeee,1

,,1

,,1, )1(

. (3.5)

Then, formula (3.3) can be replaced by

ji ji

jieji

ji

jijijji eeeee

ee , ,

,,

,

1,,,1, 22)()(

. (3.6)

Upon these transformations, the expression for the statistical sum 1Z takes the

form

ji ji

jieji

ji

jiji eeee

i eedZ , ,

,,

,

,, 22)(

1 ][

. (3.7)

The last term in the exponent in (3.7) prevents one from representing integral

(3.7) by Euler's integrals. To overcome this difficulty, we expand the last

exponential function in this formula in Taylor's series:

)......))()((6

)2(

))((2

)2()2(1(][

, ,

3

,,

2

,

2

1

,

1

,,

,,,

,

,

,

,,

ji qp

e

lk

ee

ekji

e

ji

eee

qpplkji

ekjiji

ji

ji

ji

jiji

eeee

eee

eeeedZ

.

(3.8)

Now, each term in Taylor's series can be represented by Euler's integral, which

has the form

dtetz tz

0

1)( . (3.9)

The typical integral in the series is

BeAedI . (3.10)

It is reduced to Euler's integral by the change of variables Bet :

3152 Yu. V. Bibik

A

Be

A

A

A

BeAA

B

Ae

B

BeBed

B

eB

Be

BedI

)()()(

)( 1

. (3.11)

Thus, we can reduce the series to a combination of the gamma and elementary

functions:

....)]))2((

)([]

)1)2((

)([639]

))2((

)([

)1)2((

)(

)2)2(

)(81

]))2(

)([

)3)2((

)(9(

6

)2(]

))2((

)([]

)1)2(

)([72

])2((

)([

)2)2((

)(9(

2

)2(]

))2((

)([

)1)2((

)(9)2(]

))2((

)([

637

83

72

82

89

1

e

eeZ

(3.12)

In series (3.12), the combination of the gamma and elementary functions

9]))2((

)([

can be factored out, and the remaining terms are represented by

elementary functions:

....).)

))2(

11(

639

))2(

11()

)2(

21(

81

))2(

31(

9(

6

)2(

)

))2(

11(

172

))2(

21(

9(

2

)2()

))2(

11(

1(9)2(1(]

))2((

)([

3

3

2

29

1

e

eeZ

(3.13)

It is seen from (3.13) that all terms of the series consisting of elementary functions

include the same expression

)

)2((1(

n. For the case 0 , it is equal to

unity:

1)))2(

(1( 0

n. (3.14)

A remarkable property of this expression is that it tends to a constant as .

This constant is

2)))2(

(1(

n

en

. (3.15)

Application of the statistical physics methods for the investigation… 3153

Only the linear (in ) term in (3.13) contains a single expression

)

)2(

1(1(

. The coefficients of the other terms with each power of

consist of combinations of one or several such expressions. However, the

structure of the right-hand side of formula (3.15) is such that all the combinations

of exponential functions multiplying have the same exponent for each

term of the series, which coincides with the index of the series term. This allows

us to individually sum up the terms in (3.13) as 0 and and

extrapolate the statistical sum using two values of the series. There is a more

rigorous approach. The major contribution to the terms of series (3.13) is made by

the powers of the function

)

)2(

11

1(

. The other terms exhibit similar

behavior for each n . For this reason, we expand the function

n)

)2(

11

1(

and its powers into the sum of expressions for each of which series (3.13) can be

summed separately. The desired formula has the form

2)1()

)2(

11

1(

nn eee nn . (3.16)

In (3.16), we need to determine the parameters n . Note that the right-hand side

of (3.16) provides the correct value for the power n of the function

)

)2(

11

1(

as 0 and .

It remains to require the left- and right-hand sides of formula (3.16) be equal at a

certain intermediate value of . Set 1 , and consider the case when 1 .

Then, the left-hand side of (3.16) is

nn )4

3()

)2(

11

1(

. (3.17)

Upon our transformations, formula (3.16) has the form

3)1()4

3(

nn eee nn

. (3.18)

3154 Yu. V. Bibik

Now, (3.18) implies a formula for n :

.)716.0(1

)954.01()75.0(

)716.0(1

)75.0

716.0(1

)75.0()716.0(1

)716.0()75.0(

)(1

)()4

3(

3

1

3

1

n

nn

n

n

n

n

nn

n

nn

e

e

e n

(3.19)

We transform (3.19) to a form that can be used for the summation of series (3.13):

161.0)75.0(284.0

046.0)75.0(

))284.01(1(

))046.01(1()75.0( nn

n

nn

n

ne n

. (3.20)

Formula (3.20) gives the final expression for n . Now we turn to determining the

parameter ne

. To this end, we raise the left- and right-hand sides of Eq. (3.20)

to the power and introduce the parameters )161.0ln(1 and

)75.0ln(1 . Then we obtain

11)161.0ln()75.0ln( nn eeeee n

. (3.21)

After these transformations, we can write series (3.13) as

....).))1((6

)18()1((

2

)18(

))1()(18(1(]))3((

)([

1333

3

22

2

3

19

1

1111112

11

1111

eeee

eeee

eeeeZ

(3.22)

Series (3.22) can be divided into three series of which each is just a series for the

exponential function with the corresponding exponent. The first series

corresponds to 11 nee , the second series to 3

n

e

, and the third series to

113 n

n

eee

. Each of these series is easily summed, which yields the formula

).(])3((

)([

)1(1(])3((

)([

13/21

3/211

13/21

3/2111

1818189

1818189

1

eeeee

eeeee

eeeee

eeeeeeeZ

(3.23)

Application of the statistical physics methods for the investigation… 3155

The statistical sum (3.23) interpolates the behavior of series (3.13). This fact can

be verified in the following way.

As 0 , the second and third terms in (3.23) cancel out, and only the first term

remains, which takes the form ee 18 .

As 0 , the same behavior is exhibited by formula (3.13), which confirms the

validity of our approximation.

This completes the calculation of the statistical sum for our model. Formula (3.23)

gives an approximation of the statistical sum for the generalized Lotka–Volterra

system with spatial terms. The derivation of this formula is an achievement

because the calculation of the statistical sum for the majority of models is a very

difficult task.

We managed to calculate the statistical sum analytically due to the following

trick. The last exponential function in (3.7) was expanded in Taylor's series, and

Kerner's change of variables was made in each term of this series. As a result,

each term of the series was represented by a combination of the gamma and

elementary functions. This allowed us to perform the inverse operation, i.e., sum

up the remaining series and represent it as a combination of exponential functions

multiplied by a term containing the gamma function. This is an advantage of the

proposed method because such a combination is most convenient for the further

analysis.

In the next section, we will investigate the phase transition in the model using the

expression of the statistical sum obtained above.

4. Investigation of the phase transition in the original model

The existence or nonexistence of phase transitions in the system is determined

analogously to the Lee–Yang approach [46], [70] by the position of the zeros of

the statistical sum on the complex plane. For the analysis of the position of zeros

of the statistical sum, only the expression

)(13/2

13/21

1 181818

eeeee eeeee in (3.23) is used because the

expression in brackets makes no contribution to the zeros of the statistical sum.

For this expression, a simple computer program for determining the absolute

value of this expression was developed. This absolute value was plotted using the

program Surfer. Figures 1 and 2 show the level lines of the absolute value, which

can be used to visually determine the position of zeros of the statistical sum. For

the construction of the level lines, the expression

)1(3/213/2

13/21

1 18181818 eeeeee eeee

was used, which makes it

possible to view the surface structure in the domain of the small terms for (3.23).

It is seen from Fig. 1 that the generalized Lotka–Volterra system with spatial

terms has a phase transition. This is seen from the behavior of two zeros of the

statistical sum, which are symmetric about the real axis. These two zeros are in

the right half-plane of the complex plane near the real axis. The distance from the

zeros to the real axis is 0.19. The line passing through these zeros intersects the

real axis at the point 0.034.

3156 Yu. V. Bibik

Figure 1 was constructed for the additional spatial terms )( yxpp

eed in

Hamiltonian (2.2.14) for the coefficient 1d . This implies that the phase

transition point is 0.034.

To verify this result, the level lines for the same model with the coefficient 3d

were constructed (Fig. 2). The zeros for this case are clearly seen; they testify that

the phase transition also exists in this case. The distance from two symmetric

(about the real axis) zeros from the real axis is small—it is equal to 0.13. The line

passing through these zeros intersects the real axis at the point 0.03. This implies

that the phase transition point is 0.03.

The third (rightmost) zero in Figs. 1 and 2, which lies on the real axis, has nothing

to do with phase transitions. This zero appears due to small errors in the

calculation of the approximate statistical sum (3.23) as a result of cancelling out

small terms consisting of exponential functions with large negative exponents.

Thus, the data above imply that the proposed generalization of the Lotka–Volterra

system with spatially distributed parameters has a phase transition.

0.00 0.10 0.20 0.30

-0.30

-0.20

-0.10

-0.00

0.10

0.20

0.30

Y

X

Fig. 1

Roots of statistical sum for d=1

Application of the statistical physics methods for the investigation… 3157

0.00 0.10 0.20 0.30

-0.30

-0.20

-0.10

-0.00

0.10

0.20

0.30

Y

X

Fig. 2

Roots of statistical sum for d=3

5. Conclusions

In statistical physics, there are a lot of techniques for the analysis of phase

transitions. For each problem, an appropriate most convenient approach can be

found. The Lee–Yang method [46], [70] makes it possible to investigate the phase

transitions judging by the position of the zeros of the statistical sum on the

complex plane. To apply this method, one must know how to calculate the

statistical sum, which is a difficult problem. In this paper, the case when this sum

can be calculated was considered. A generalization of the Lotka–Volterra system

that takes into account the spatial variables was studied. The Lotka–Volterra

system has a specific Hamiltonian that allows one to calculate the statistical sum

in terms of gamma and elementary functions. A special choice of the spatial terms

3158 Yu. V. Bibik

makes it possible to preserve this property for the generalized system, which was

done in the present paper.

The addition of spatial dependence of the population of species to the classical

Lotka–Volterra system makes it possible to study more complex behavior

patterns, including the emergence of phase transitions.

The Hamiltonian equations (2.2.15), (2.2.16) for the system under examination

are nothing more nor less than nonlinear wave equations that have soliton-like

solutions. In this system, as in all systems with phase transitions, there are two

counteracting factors. The first factor "wants" to order the system in the form of

soliton-like solutions. The second factor, in the form of temperature, strives to

destroy the ordering. At a certain relation of these factors, a phase transition

occurs.

The proper choice of the functions that take into account the spatial dependence of

the population of species and their ultimate critical population made it possible to

perform all the calculations analytically. As a result, the position of zeros of the

statistical sum on the complex plane was analyzed analytically, and the condition

of the occurrence of the phase transition was obtained judging by this position.

It seems that the study should be continued to obtain a greater number of zeros

using a grid with a greater number of points. Since this considerably complicates

the mathematical calculations, this issue is left out of the scope of the present

paper.

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Received: October 12, 2016; Published: November 28, 2016